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Let F (x) = f(x) + f ((1)/(x)), where f ...

Let `F (x) = f(x) + f ((1)/(x)),` where `f (x) = int _(1) ^(x ) (log t)/(1+t) dt.` Then F (e) equals

A

`1//2`

B

0

C

1

D

2

Text Solution

Verified by Experts

The correct Answer is:
A
`F(e)=f(e)+f((1)/(e))`
`rArrF(e)=int_(1)^(e)(logt)/(1+t)dt+int_(1)^(1//e)(logt)/(1+t)dt[becausef(x)=int_(1)^(x)(logt)/(1+t)dt]`
On putting t `=(1)/(t)` in second integration , we get
`F(e)=int_(1)^(e)(logt)/(1+t)dt+int_(1)^(e)(logt)/(t(1+t))`
`=int_(1)^(e)(logt)/(t)dt=[((logt)^(2))/(2)]_(1)^(e)`
`=(1)/(2)[(loge)^(2)-(log1)^(2)]=(1)/(2)`
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MHTCET PREVIOUS YEAR PAPERS AND PRACTICE PAPERS-DEFINITE INTEGRALS-PRACTICE EXERCISE (Exercise 2) (MISCELLANEOUS PROBLEMS)
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  3. Let F (x) = f(x) + f ((1)/(x)), where f (x) = int (1) ^(x ) (log t)/(1...

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  4. If 2f(x) - 3 f(1//x) = x," then " int(1)^(2) f(x) dx is equal to

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  5. int (0)^(2pi) sin^(6) x cos^(5) x dx is equal to

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  6. int (-3pi//2)^(-pi//2) [ ( x + pi)^(3) + cos^(2) x ] dx is equalt to

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  7. int (0)^(3) (3x+1)/(x^(2)+9) dx =

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  8. underset(n to oo)lim underset(r=1)overset(n)sum(1)/(n)e^(r//n) is

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  9. int (-1//2)^(1//2) cos x log ((1+x)/(1-x)) dx = k log 2 , then k equ...

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  10. The value of the integral int (0)^(pi//2) sin ^(5) x dx is

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  11. If int (0)^(x^(2)) f(t) dt = x cos pix , then the value of f(4) is

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  12. If f(x) is differentiable and int0^(t^2)xf(x)dx=2/5t^5, then f(4/(25))...

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  13. The value of int (0)^(pi//2) sin ^(8) x dx is

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  14. The value of overset(pi)underset(-pi)int(1-x^(2)) sin x cos^(2) x" dx"...

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  15. int (0)^(1) (xdx)/([x + sqrt(1-x^(2))]sqrt(1-x^(2))) is equal to

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  16. The value of int(0)^(pi)(Sigma(r=0)^(3)a(r)cos^(3-r)x sin^(r)x)dx depe...

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  17. The value of int (-1)^(1) x|x| dx is

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  18. The value of integral int (1//pi)^(2//pi)(sin(1/x))/(x^(2))dx=

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  19. The value of int (1)^(e^(2)) (dx)/(x(1+ log x)^(2) ) is

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  20. int(pi//3)^(pi//2) (sqrt(1+cosx))/((1-cosx)^(5//2)) dx

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